ComplexEigenSolver.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009 Claire Maurice
// Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_COMPLEX_EIGEN_SOLVER_H
#define EIGEN_COMPLEX_EIGEN_SOLVER_H

#include "./ComplexSchur.h"

// IWYU pragma: private
#include "./InternalHeaderCheck.h"

namespace Eigen {

template <typename MatrixType_>
class ComplexEigenSolver {
 public:
  typedef MatrixType_ MatrixType;

  enum {
    RowsAtCompileTime = MatrixType::RowsAtCompileTime,
    ColsAtCompileTime = MatrixType::ColsAtCompileTime,
    Options = internal::traits<MatrixType>::Options,
    MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
    MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
  };

  typedef typename MatrixType::Scalar Scalar;
  typedef typename NumTraits<Scalar>::Real RealScalar;
  typedef Eigen::Index Index;  

  typedef internal::make_complex_t<Scalar> ComplexScalar;

  typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & (~RowMajor), MaxColsAtCompileTime, 1> EigenvalueType;

  typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime,
                 MaxColsAtCompileTime>
      EigenvectorType;

  ComplexEigenSolver()
      : m_eivec(), m_eivalues(), m_schur(), m_isInitialized(false), m_eigenvectorsOk(false), m_matX() {}

  explicit ComplexEigenSolver(Index size)
      : m_eivec(size, size),
        m_eivalues(size),
        m_schur(size),
        m_isInitialized(false),
        m_eigenvectorsOk(false),
        m_matX(size, size) {}

  template <typename InputType>
  explicit ComplexEigenSolver(const EigenBase<InputType>& matrix, bool computeEigenvectors = true)
      : m_eivec(matrix.rows(), matrix.cols()),
        m_eivalues(matrix.cols()),
        m_schur(matrix.rows()),
        m_isInitialized(false),
        m_eigenvectorsOk(false),
        m_matX(matrix.rows(), matrix.cols()) {
    compute(matrix.derived(), computeEigenvectors);
  }

  const EigenvectorType& eigenvectors() const {
    eigen_assert(m_isInitialized && "ComplexEigenSolver is not initialized.");
    eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
    return m_eivec;
  }

  const EigenvalueType& eigenvalues() const {
    eigen_assert(m_isInitialized && "ComplexEigenSolver is not initialized.");
    return m_eivalues;
  }

  template <typename InputType>
  ComplexEigenSolver& compute(const EigenBase<InputType>& matrix, bool computeEigenvectors = true);

  ComputationInfo info() const {
    eigen_assert(m_isInitialized && "ComplexEigenSolver is not initialized.");
    return m_schur.info();
  }

  ComplexEigenSolver& setMaxIterations(Index maxIters) {
    m_schur.setMaxIterations(maxIters);
    return *this;
  }

  Index getMaxIterations() { return m_schur.getMaxIterations(); }

 protected:
  EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)

  EigenvectorType m_eivec;
  EigenvalueType m_eivalues;
  ComplexSchur<MatrixType> m_schur;
  bool m_isInitialized;
  bool m_eigenvectorsOk;
  EigenvectorType m_matX;

 private:
  void doComputeEigenvectors(RealScalar matrixnorm);
  void sortEigenvalues(bool computeEigenvectors);
};

template <typename MatrixType>
template <typename InputType>
ComplexEigenSolver<MatrixType>& ComplexEigenSolver<MatrixType>::compute(const EigenBase<InputType>& matrix,
                                                                        bool computeEigenvectors) {
  // this code is inspired from Jampack
  eigen_assert(matrix.cols() == matrix.rows());

  // Do a complex Schur decomposition, A = U T U^*
  // The eigenvalues are on the diagonal of T.
  m_schur.compute(matrix.derived(), computeEigenvectors);

  if (m_schur.info() == Success) {
    m_eivalues = m_schur.matrixT().diagonal();
    if (computeEigenvectors) doComputeEigenvectors(m_schur.matrixT().norm());
    sortEigenvalues(computeEigenvectors);
  }

  m_isInitialized = true;
  m_eigenvectorsOk = computeEigenvectors;
  return *this;
}

template <typename MatrixType>
void ComplexEigenSolver<MatrixType>::doComputeEigenvectors(RealScalar matrixnorm) {
  const Index n = m_eivalues.size();

  matrixnorm = numext::maxi(matrixnorm, (std::numeric_limits<RealScalar>::min)());

  // Compute X such that T = X D X^(-1), where D is the diagonal of T.
  // The matrix X is unit triangular.
  m_matX = EigenvectorType::Zero(n, n);
  for (Index k = n - 1; k >= 0; k--) {
    m_matX.coeffRef(k, k) = ComplexScalar(1.0, 0.0);
    // Compute X(i,k) using the (i,k) entry of the equation X T = D X
    for (Index i = k - 1; i >= 0; i--) {
      m_matX.coeffRef(i, k) = -m_schur.matrixT().coeff(i, k);
      if (k - i - 1 > 0)
        m_matX.coeffRef(i, k) -=
            (m_schur.matrixT().row(i).segment(i + 1, k - i - 1) * m_matX.col(k).segment(i + 1, k - i - 1)).value();
      ComplexScalar z = m_schur.matrixT().coeff(i, i) - m_schur.matrixT().coeff(k, k);
      if (z == ComplexScalar(0)) {
        // If the i-th and k-th eigenvalue are equal, then z equals 0.
        // Use a small value instead, to prevent division by zero.
        numext::real_ref(z) = NumTraits<RealScalar>::epsilon() * matrixnorm;
      }
      m_matX.coeffRef(i, k) = m_matX.coeff(i, k) / z;
    }
  }

  // Compute V as V = U X; now A = U T U^* = U X D X^(-1) U^* = V D V^(-1)
  m_eivec.noalias() = m_schur.matrixU() * m_matX;
  // .. and normalize the eigenvectors
  for (Index k = 0; k < n; k++) {
    m_eivec.col(k).stableNormalize();
  }
}

template <typename MatrixType>
void ComplexEigenSolver<MatrixType>::sortEigenvalues(bool computeEigenvectors) {
  const Index n = m_eivalues.size();
  for (Index i = 0; i < n; i++) {
    Index k;
    m_eivalues.cwiseAbs().tail(n - i).minCoeff(&k);
    if (k != 0) {
      k += i;
      std::swap(m_eivalues[k], m_eivalues[i]);
      if (computeEigenvectors) m_eivec.col(i).swap(m_eivec.col(k));
    }
  }
}

}  // end namespace Eigen

#endif  // EIGEN_COMPLEX_EIGEN_SOLVER_H